On links in $S_{g} \times S^{1}$ and its invariants

TL;DR

This paper explores knots in the 3-manifold $S_{g} imes S^{1}$, introducing basic notions, diagrammatic moves, and invariants based on the rotation information, extending knot theory in thickened surfaces.

Contribution

It introduces a framework for studying knots in $S_{g} imes S^{1}$, including diagrams, moves, and invariants derived from rotation data, expanding the understanding of virtual knots.

Findings

Defined diagrams and moves for knots in $S_{g} imes S^{1}$

Developed invariants using rotation information

Connected geometric meaning to rotation data

Abstract

A virtual knot, which is one of generalizations of knots in $\mathbb{R}^{3}$ (or $S^{3}$), is, roughly speaking, an embedded circle in thickened surface $S_{g} \times I$. In this talk we will discuss about knots in 3 dimensional $S_{g} \times S^{1}$. We introduce basic notions for knots in $S_{g} \times S^{1}$, for example, diagrams, moves for diagrams and so on. For knots in $S_{g} \times S^{1}$ technically we lose over/under information, but we will have information how many times the knot rotates along $S^{1}$. We will discuss the geometric meaning of the rotating information and how to construct invariants by using the "rotating" information.